Exercise 8, § 4

(a) Let $E$ be a distributive lattice (§ 1, Exercise 16). If $a$ is irreducible in $E$ (Exercise 7), show that the relation $a\leq\sup(x,y)$ implies $a\leq x$ or $a\leq y$.

(b) Let $E$ be a finite distributive lattice and let $J$ be the set of its irreducible elements, ordered by the induced ordering. Show that the isomorphism $x\to S(x)$ of $E$ onto a subset of $\mathfrak{P}(J)$ defined in Exercise 7 (b) is such that $S(\sup(x,y))=S(x)\cup S(y)$. Deduce that if $J^*$ is the ordered set obtained by endowing $J$ with the opposite ordering, then $E$ is isomorphic to the set $\mathscr{A}_0(J^*,I)$ of increasing mappings of $J^*$ into $I=\{0,1\}$ (§ 1, Exercise 6).

(c) With the hypotheses of (b), let $P$ be the set of elements of $J$ other then the least element of $E$. For each $x\in E$, let $y_1,\ldots,y_k$ be the distinct minimal elements of the interval $]x,\to[$ in $E$; for each index $i$ let $q_i$ be an element of $P$ such that $q_i\notin S(x)$ and $q_i\in S(y_i)$. Show that no two of the elements $q_1,\ldots,q_k$ are comparable.

(d) Conversely, let $q_1$, ..., $q_k$ be $k$ elements of P, no two of which are comparable. Let $u = \sup{(q_1, \ldots, q_k)}$ and let

Show that $v_i < u$ for $1 \leqslant i \leqslant k$. Let $x = \inf{(v_1, \ldots, v_k)}$ and let

Show that $x < y_i$ for each index $i$, and deduce that the interval $]x, \rightarrow[$ has at least $k$ distinct minimal elements.